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Polynomial multiplication is a foundational concept in algebra that involves multiplying two or more polynomials together. In the given exercise, we have an expression

• Single monomial: This is a single polynomial term, such as \(-3b\).
• Polynomial: Here, this is a binomial inside the parentheses \(2b^2 - b\).

To multiply these, we apply each term of the monomial to every term in the polynomial inside the parentheses. This comprehensive multiplication of individual terms is the essence of polynomial multiplication. To solidify understanding, remember:

• Each term in the polynomial is treated individually.
• We multiply the coefficients (numbers) and add the exponents for the same bases.

The distributive property is a powerful tool when working with algebraic expressions, especially for polynomial multiplication. It instructs us to "distribute" the outer term across each term within a parenthesis. Here’s how it works in our example:

• Distribute \(-3b\) to \(2b^2\): Multiply the coefficients: \(-3 \times 2 = -6\). Then, add the exponents of \(b\): \(1 + 2 = 3\), resulting in \(-6b^3\).
• Distribute \(-3b\) to \(-b\): Multiply the coefficients: \(-3 \times -1 = 3\). Add the exponents of \(b\): \(1 + 1 = 2\), yielding \(3b^2\).

The distributive property ensures that each term is multiplied appropriately and every product is accounted for in the final result.

A step-by-step solution is invaluable for understanding the complexity of polynomial multiplication. Let's break down our exercise further:1. **Apply the Distributive Property**
Begin the process by focusing on the first term, \(-3b\), and multiply it with each term in the binomial \(2b^2 - b\). Start with \(2b^2\), then move to \(-b\). 2. **Calculate Each Product**
- For \(-3b \times 2b^2\), result is \(-6b^3\). - For \(-3b \times (-b)\), result is \(3b^2\).

3. **Combine Like Terms**
The last step is to combine all distributed products into a single expression. It results in: \[-6b^3 + 3b^2\]This approach breaks down the algebra problem into manageable parts, making it easier to handle and understand the entire multiplication process from start to finish.